The power generated by an electrical circuit (in watts) as a function of its current $x$ (in amperes) is modeled by: $P(x)=-12x^2+120x$ What is the maximum power generated by the circuit?
Solution: The circuit's power is modeled by a quadratic function, whose graph is a parabola. The maximum power is reached at the vertex. So in order to find the maximum power, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $P(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} P(x)&=0 \\\\ -12x^2+120x&=0 \\\\ x^2-10x&=0 \\\\ x(x-10)&=0 \\\\ \swarrow &\searrow \\\\ x=0\text{ or }&x-10=0 \\\\ x={0}\text{ or }&x={10} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({10})}{2}=\dfrac{10}{2}= 5$ The vertex's $x$ -coordinate is ${5}$. Now let's find $P({5})$ : $\begin{aligned} P({5})&=-12({5})^2+120({5}) \\\\ &=-300+600 \\\\ &=300 \end{aligned}$ In conclusion, the circuit generates a maximum power of $300$ watts.